Linear symmetric determinantal hypersurfaces

Abstract

The question which equations of hypersurfaces in the complex projective space can be expressed as the determinant of a matrix whose entries are linear forms is classical. In 1844 Hesse proved that a smooth plane cubic has three essentially different linear symmetric representations [He]. Dixon showed in 1904 that for smooth plane curves linear symmetric determinantal representations correspond to ineffective theta–characteristics, i.e., ineffective divisor classes whose double is the canonical divisor [Di]. Barth proved the corresponding statement for singular plane curves [B]. The general case for any hypersurface was treated by Catanese [C], Meyer–Brandis [M–B], and Beauville [Be].